By P. K. Jain, Ahmed Khalid
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Extra info for A Textbook Of Analytical Geometry Of Two Dimensions
The complete projective topoy W for complete locally convex topological vector spaces logical tensor product V ˝ y W ! X is defined by a similar universal property: continuous linear maps V ˝ correspond to jointly continuous bilinear maps V W ! X . This tensor product is defined by Alexander Grothendieck in . Let V and W be complete locally convex topological vector spaces. W / ! W / ! W / ! W / ! V ˝ are bounded. The issue is when these maps are bornological isomorphisms. We can only expect positive results for special topological vector spaces like Fréchet spaces.
1 Subspaces and quotients Let V be a bornological vector space and let W Â V be a subspace. 57. V /g: We always equip subspaces with this bornology unless otherwise specified. This bornology is characterised by the property that a linear map f W X ! W is bounded if and only if it is bounded as a map to V . An analogous assertion holds for uniformly bounded sets of linear maps. X; V /. 58. Let V be a locally convex topological vector space and let W Â V be a subspace, endowed with the subspace topology.
This shows that Cpt C ! M; V / D C ! V / . Literally the same argument works for von Neumann bounded subsets instead of precompact subsets, so that we also get vN C ! M; V / D C ! V / . U; V / with the space of continuous functions X ! V that vanish outside U . X; V / of compactly supported continuous functions X ! U; V / for some relatively compact open subset U Â X . M; V / for k 2 N [ f1g is defined similarly. 37. xn / in A. These subsets satisfy the axioms for a topology, which we call the bornological topology on V .